Rational Surface Singularities Research Articles

Vanishing of Tors of absolute integral closures in equicharacteristic zero

We show that R R is regular if T o r i R ( R + , k ) = 0 Tor_{i}^{R}(R^{+},k) = 0 for some i ≥ 1 i\geq 1 assuming further that R R is a N \mathbb {N} -graded ring of dimension 2 2 finitely generated over an algebraically closed equicharacteristic zero field k k . This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R + R^{+} to deduce properties of the noetherian ring R R and rational surface singularities. Moreover we observe that R + R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m m -adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i ≪ d i m ( R ) i \ll dim(R) . We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding ‘fpqc analogues’ of splinters.

Unexpected Stein fillings, rational surface singularities and plane curve arrangements

Unexpected Stein fillings, rational surface singularities and plane curve arrangements

Cohomological characterizations of smoothness and the case of rational surface singularities

We establish characterizations of smoothness for a complex affine algebraic variety at a given Cohen-Macaulay point. Our main result treats the case of surfaces with (at most) rational singularities, by a technique that requires the vanishing of suitable Ext modules rather than of sheaf cohomology groups. We also prove results in arbitrary dimension which, in addition to smoothness, detect curves and surfaces as a global feature.

Delta invariant of curves on rational surfaces I. An analytic approach

We prove that if [Formula: see text] is a reduced curve germ on a rational surface singularity [Formula: see text] then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair [Formula: see text]. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann–Roch formula, valid for projective normal surfaces, introduced by Blache.

Covers of rational double points in mixed characteristic

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.

Weighted Projective Lines and Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.

The embedded Nash problem of birational models of rational triple singularities

We consider the question whether one can construct an embedded resolution of singularities of a singular variety X⊂An from the data of the irreducible components of the spaces of jets (of X) centered at the singular locus of X. We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.

Minimal surface singularities are Lipschitz normally embedded

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.

A characterization of Lipschitz normally embedded surface singularities

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), i.e., to have bilipschitz equivalent outer and inner metrics. In a partner paper [15] we apply it to prove that rational surface singularities are LNE if and only if they are minimal.

Deformations of rational surface singularities and reflexive modules with an application to flops

Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.

Deformations of algebras defined by tilting bundles

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to C⁎-equivariant deformations over schemes with a good C⁎-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the C⁎-equivariant case.We apply our results to rational surface singularities, generalising the deformed preprojective algebras of Crawley-Boevey and Holland, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras.

Quiver GIT for varieties with tilting bundles

In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra A:=mathrm{End}_{X}(T)^{mathrm{op}}. We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for A=mathrm{End}_{X}(T)^{mathrm{op}} then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities.

The number of equisingular moduli of a rational surface singularity

We consider a conjectured topological inequality for the number of equisingular moduli of a rational surface singularity, and prove it in some natural special cases. When the resolution dual graph is sufficiently negative (in a precise sense), we verify the inequality via an easy cohomological vanishing theorem, which implies that this number is computed simply from the graph (Theorem 3.10). To consider an important and less restrictive meaning of sufficiently negative requires a much more difficult hard vanishing theorem (Theorem 4.5), which is false in characteristic p. Theorem 7.9 verifies the conjectured inequality in this more general situation. As a corollary, we classify in characteristic p all taut singularities with reduced fundamental cycle (Theorem 9.2).

Toric Ideals of Simple Surface Singularities

In this paper, we study a class of toric ideals obtained by using some geometric data of ADE trees which are the minimal resolution graphs of rational surface singularities. We compute explicit Gr\"obner bases for these toric ideals that are also minimal generating sets consisting of large number of binomials of degree $\leq 4$. In particular, they give rise to squarefree initial ideals as well.

Frobenius categories, Gorenstein algebras and rational surface singularities

AbstractWe give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

On the classification of rational surface singularities

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.

Arcs and wedges on rational surface singularities

Given a rational surface singularity (S,P0) over an algebraically closed field of characteristic 0, we prove that its minimal desingularization satisfies the property of lifting wedges centered at those stable points Pα of the space of arcs S∞ which correspond to the essential divisorial valuations. This proves the Nash problem for rational surface singularities and, more generally, reduces the Nash problem for surfaces to quasirational normal singularities which are not rational. In positive characteristic, we give counterexamples to the k-wedge lifting problem for the surface singularity x3+y5+z2=0.

Milnor open books of links of some rational surface singularities

We describe Milnor open books and Legendrian surgery diagrams for canon- ical contact structures of links of some rational surface si ngularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Mil- nor norm) is strictly greater than the support genus (resp. s upport norm) of the canonical contact structure, for each member of this family.

Arcs and wedges on rational surface singularities

Let ( S , P 0 ) be a rational surface singularity over an algebraically closed field k of characteristic 0, let ν α be an essential divisorial valuation over ( S , P 0 ) , and P α the stable point of the space of arcs S ∞ corresponding to ν α . We prove that any wedge centered at P α lifts to the minimal desingularization. This proves the Nash problem for rational surface singularities, and reduces the Nash problem for surfaces to quasirational normal singularities which are not rational. In positive characteristic, we give a counterexample to the k-wedge lifting problem for a surface for which the Nash map is bijective.

On rational homology disk smoothings of valency 4 surface singularities

Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the three classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in a previous paper; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class. We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz, is evidence for a general Conjecture: The only complex surface singularities with a QHD smoothing are the (known) weighted homogeneous examples.